Multiplying Positive and Negative Numbers

Date: 10/27/2003 at 18:15:17
From: kirsten
Subject: math
In a problem like
-3 x 7 x 4 = ?
what do I do with the negative numbers?

Date: 10/27/2003 at 22:26:37
From: Doctor Ian
Subject: Re: math
Hi Kirsten,
Here's the basic idea. A negative number is the same as -1 times the
corresponding positive number. For example, -3 is just -1*3, -18 is
just -1*18, and so on. (I'm using '*' to represent multiplication.)
How does this help? Well, suppose we have a bunch of positive and
negative numbers being multiplied, like
3 * -4 * 5 * -2 * -7 = ?
We can start by replacing the negatives with -1 times the
corresponding positives:
3 * -1 * 4 * 5 * -1 * 2 * -1 * 7 = ?
How does this help us? Well, since multiplication is commutative,
order doesn't matter; so we can move all the -1's out to one side:
-1 * -1 * -1 * 3 * 4 * 5 * 2 * 7 = ?
The next thing to know is that -1 * -1 = 1. So whenever we have a
pair of -1's multiplied together, we can replace them with a 1... and
since multiplying by 1 doesn't change anything, we can just eliminate
the 1:
(-1 * -1) * -1 * 3 * 4 * 5 * 2 * 7 = ?
(1) * -1 * 3 * 4 * 5 * 2 * 7 = ?
-1 * 3 * 4 * 5 * 2 * 7 = ?
Now, we just have to multiply the other numbers together, and since we
still have a -1 hanging around, the sign of the whole thing is negative.
Now, there is a shortcut we can use, but it's only worth doing if we
realize _why_ it works. Instead of dragging the -1's out front to
kill each other off, we can just kill of pairs of signs in place. For
example, given
3 * -4 * 5 * -2 * -7 = ?
we could kill off a pair of signs:
3 * -4 * 5 * -2 * -7 = ?
^ ^
|________|_________________ these signs cancel
3 * 4 * 5 * 2 * -7 = ?
Now that there's only one sign left, we can bring it out front:
-(3 * 4 * 5 * 2 * 7) = ?
Does this help?
- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/